Question: $f(x, y, z) = \left( -y + z, -z + x, -x + y \right)$ What is the divergence of $f$ at $(1, 2, 3)$ ?
The formula for divergence in three dimensions is $\text{div}(f) = \dfrac{\partial P}{\partial x} + \dfrac{\partial Q}{\partial y} + \dfrac{\partial R}{\partial z}$, where $P$ is the $x$ -component of $f$, $Q$ is the $y$ -component, and $R$ is the $z$ -component. Let's differentiate! $\begin{aligned} \dfrac{\partial P}{\partial x} &= \dfrac{\partial}{\partial x} \left[ -y + z \right] \\ \\ &= 0 \\ \\ \dfrac{\partial Q}{\partial y} &= \dfrac{\partial}{\partial y} \left[ -z + x \right] \\ \\ &= 0 \\ \\ \dfrac{\partial R}{\partial z} &= \dfrac{\partial}{\partial z} \left[ -x + y \right] \\ \\ &= 0 \end{aligned}$ Adding the three partial derivatives, $\text{div}(f) = 0$. The divergence of $f$ at $(1, 2, 3)$ is $0$.